Connecting Topological Hopf Singularities
ROBERT HARDT – TRISTAN RIVI `ERE
Abstract. Smooth maps between Riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis- Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map.
The relevant algebraic object here isπ2(S2)which provides both the obstruction to strong approximation by smooth maps and the topological structure to the bubbling set and the singular set. With higher homotopy groups, new phenomena occur. Forπ3(S2)and the related case of finite 3-energy maps from the 4 Ball to the 2 sphere, there are examples with bubbled objects that no longer have finite mass. We define a new object, a scan, which generalizes a current but still occurs naturally in bubbling while automatically providing the topological connection between the singularities of the limit map. The bubbled scans, which are found via a new compactness theorem, again enjoy a representation using a finite measure 1 rectifiable set and an integer density function which is now however onlyL3/4 (rather thanL1) integrable.
Mathematics Subject Classification (2000):58D15 (primary), 58E20, 49Q15 (secondary)
0. – Introduction
There are many interesting questions and works concerning the relation between the topology of Riemannian manifolds M and N and the structure of the various Sobolev spaces Ws,p(M,N) of maps between them. For example, the space W1,p(M,Sp) of finite p energy maps to the p sphere and issues concerning the possible approximability by smooth maps have been well-studied by F. Bethuel [Be1] and others using the notion of topological degree, which is associated withπp(Sp). For dim :M < p, these Sobolev maps are automatically Research partially supported by NSF grant DMS-0072486.
Pervenuto alla Redazione il 26 ottobre 2001 e in forma definitiva il 7 febbraio 2003.
continuous. In the critical dimension dimM = p, one has the phenomenon of bubbling whereby a weakly convergent sequence of smooth maps may, in the limit, drop energy and topological degree and produce, in a suitable space, auxiliary objects (bubbles) accounting for topological changes near a finite set of points. In case dimM > p, the limiting map itself may have essential topological singularities, detected by degree, which are topologically connected by a bubbling set of dimension dimM −[p]. These are particularly well- understood for dimM =3,p=2 [HL1], [BCL], [Be2], [BBC], [GMS1] where, for example, the bubbling set carries a 1 dimensional finite mass rectifiable current whose boundary is the topological singularities of the limit map.
In general, the homotopy groupπp(N)should be used to study the Sobolev spaces W1,p(M,N). In the present paper we work with the Hopf invari- ant, which is associated with π3(S2), to understand spaces W1,3(M,S2) where di m M =4. We discover some new phenomena. Examples in Section 2.5 show that now the bubbled object can possibly have infinite one dimensional mass and that the singularities that appear in weak limits of sequences of smooth maps may possibly not bound any finite mass current. We define in Section 2 a new object, a scan, which generalizes a current but still occurs naturally in bubbling while automatically providing the topological connection between the singularities of the limit map. The bubbled scans, which are found in Section 6 via a new compactness theorem, again enjoy a representation in Section 7 using a finite measure 1 rectifiable set and an integer density function which is now however only L3/4 integrable (rather than L1 integrable).
Background.
With the target manifold N viewed as isometrically embedded in a Eu- clidean space Rk, one may define, for positive numbers p and s, with p≥1, the Sobolev space
Ws,p(M,N)= {u∈Ws,p(M,Rk):u(x)∈N for a.e. :x ∈M},
where the vector space Ws,p(M,Rk) is obtained from Wlocs,p(RdimM,Rk) using local coordinate charts for M.
In contrast to the vector-space case N = Rk, some Sobolev maps u ∈ Ws,p(M,N) do not admit approximation by a sequence of smooth maps un∈ C∞(M,N)in the strong or even in the weak Ws,p(M,N)topologies. Questions about density ofC∞(M,N)in Ws,p(M,N)arise naturally for example from the study of variational problems among manifolds such as with harmonic maps, etc. [SU], [W1], [W2], [HL1], [BZ], [Be1], [BCL], [BBC], [GMS2]. Recently the path-connectness of W1,p(M,N)has been studied in [BL], [HgL1], [HgL2].
As a first approach to the notion of topological singularity, with M being the open unit ballBm inRm, we may define thetopological singular setof a map u∈Ws,p(Bm,N)as the largest open subset of Bm on which u is Ws,p strongly approximable. The obstruction to the strong approximation is characterized by the appearance, locally around the singularities of u, of nonzero elements of
πk(N) where k = [sp], the integer part of sp. For example, the fact [SU]
that the map u : B3 → S2, u(x) = x/|x|, is not strongly approximable in W1,p(B3,S2) (for2 ≤ p ≤ 3) by regular maps is due to the realization of a nonzero element of π2(S2) on spheres surrounding the singularity 0. More generally, one has, for s=1,
Theorem 0.1 [SU], [BZ], [Be1]. The space C∞(Bm,N)is strongly dense in W1,p(Bm,N)if and only if
p≥m or π[p](N)=0.
As recently observed by F. Hang and F.H.Lin [HgL1], [HgL2], this sufficient condition for strong density does not extend to an arbitrary domain. The map v from CP3 to CP2, defined in homogeneous coordinates by v[z1,z2,z3,z4]= [z1,z2,z3], has a singularity at a = [0,0,0,1] and admits no global strong approximation by smooth maps. While the above theorem gives the existence of local obstructions due only to π[p](N), this counterexample illustrates global obstruction. Here the singularity is delocalized in the sense that one may very well approximate the above vstrongly in W1,3(CP3,CP2)by maps smooth in a fixed neighborhood of a because, one may, with arbitrarily small energy, “order the globally essential singularity to reappear somewhere else.” By contrast, the local obstructions are fixed in space: it is impossible to strongly approximate u(x)=x/|x| in W1,2(B3,S2) by a sequence of maps smooth in a fixed neigh- borhood of the limit singularity 0. This is the phenomenon that we wish to study here, and we will thus restrict especially to the domain M = Bm. We also restrict to the case s=1 although certain results below extend to fractional Sobolev spaces (see [Be3], [Ri2]).
Wheneverπ[p](N)=0,C∞(M,N)is too small to “cover by strong density”
all of W1,p(Bm,N), and one uses the following larger space R∞,p(Bm,N)= {u∈C∞(Bm\A,N)∩W1,p: A is an n−[p]−1
dimensional smooth cell complex and [u|SxA]=0 inπ[p](N) for a.a. :x∈ A} where u|SxA is the restriction of u to any sufficiently small [p] dimensional sphere normal to A at x. One then has the following:
Theorem 0.2[Be1]. For[p]>1, R∞,p(Bm,N)W1,p =W1,p(Bm,N). For example, R∞,2(B3,S2) consists of maps u ∈ W1,2(B3,S2) that are smooth away from a finite set Aand whose restriction to any small spheres about a point of K has nonzero degree. In particular, u(x)=x/|x| ∈R∞,2(B3,S2)\ C∞(B3,S2)W1,2.
Definition. For u∈ R∞,p(Bm,N) one defines the topological singularity of u, Singtopu as the flat π[p](N) chain obtained from the singular set K by assigning to each point x ∈ K the representative of [u|SxK] in π[p](N). Flat G chains are defined in [Fl] (see also [F], [GMS2], [W3]). For example the
topological singularity of an element of R∞,1(B3,RP2) is a sum of disjoint unoriented curves in B3 because π1(RP2)=Z2.
The general question motivating the present paper is the following. Being given a sequenceun in R∞,p(Bm,N), converging strongly in W1,p(Bm)to a limit u, may one experience some convergence of the flatπ[p](N) chains Singtopun to a limit “object Singtopu” which will depend only on u and will characterize the approximability of u by smooth maps in W1,p in particular, if Singtopu =0, then u∈C∞(Bm,N)W1,p.
As we will see below, the understanding of the behavior of the topological singularities of maps strongly convergent in W1,p is linked to the problem of weak sequential density.
A well-understood case: p(Sp).
One may consider W1,p(Bm,Sp) where m > p are positive integers. For simplicity we treat the specific case p = 2,m = 3, keeping in mind that the set of results below extends to the general case.
So consider un ∈ R∞,2(B3, S2) strongly convergent in W1,2 to u ∈ W1,2(B3,S2). Then Singtopun is simply a finite sum of integer multiples of point masses a∈Anmn,a[[a]]. It isn’t difficult to see that these distributions are characterized by the formula
a∈An
mn,a[[a]] = ∗d u#n ωS2
2π
where ωS2 is the volume form of S2. From the strong W1,2 convergence ofun
one deduces without difficulty the convergence Singtopun = ∗d u#n
ωS2
2π
→ ∗d u# ωS2
2π
in D(B3) ,
independent of un, which is the desired topological singularity of u. On has also the
Theorem 0.3 [Be2]. d u#ωS2 =0 ⇐⇒ u∈C∞(B3,S2)W1,2.
The relation between the topological singularities and theweak convergence of smooth maps is understood by means of the following
Theorem 0.4 [Be2], [BCL], [GMS2]. For u ∈W1,2(B3,S2), there exists a1 dimensional rectifiable current such that∂I = ∗d(u#ωS2)and
8πM(I) ≤
B3|∇u|2 whereM(I)is the mass(or length)of I .
In order to approximate a map u having d(u#ω) = 0 weakly by smooth maps, it suffices to “withdraw” the topological singularities using a finite amount of W1,2 energy. This is accomplished (see [Be2]) by inserting some coverings of S2 along I which, by the above estimate, costs exactly 8πM(I)+ (with being arbitrarily small). One thus obtains the sequential weak density:
Theorem 0.5 [Be2]. For any u in W1,2(B3,S2)there exists uninC∞(B3,S2) which converge weakly to u in W1,2.
Finally there is another elegant method of characterizing the topological singularity of a map in W1,2(B3,S2), constant on ∂B3, as being the “holes” of its graph.
Theorem 0.6 [GMS1]. For any sequence of maps un ∈ C∞(B3,S2)that is W1,2weakly convergent to u ∈W1,2(B3,S2), there exists a subsequence un and a one dimensional rectifiable current I , so that one has the weak convergence of the three dimensional rectifiable currents
Graph(un) → Graph(u) + I ×[[S2]]. Moreover,
∂Graph(u) = ∂I ×[[S2]] with ∂I = ∗d u# ωS2
2π
.
An example of a more complex case: π3(S2).
This is the first case of an infinite homotopy group of spheres which is different from πp(Sp). Thus in the present paper we take N = S2, m = 4 and p = 3 and work with the Sobolev space W1,3(B4,S2). The space R∞,3(B4,S2) which now consists of maps in W1,3(B4,S2) which are smooth outside a finite set of points and realize the nontrivial elements ofπ3(S3)Zon sufficiently small spheres centered at these points. Once again the topological singularity is identified with finite atomic measures having integer multiplicities.
Also R∞,3(B4,S2) is again strongly dense in W1,3(B4,S2) [Be1]. Criteria for a given map in W1,3(B4,S2) to be strongly approximable by smooth maps have been obtained by Zhou [Z] and Isobe [I1] who also considered [I2] gap phenomena [HL1] for this space. Being given a sequence un of elements of R∞,3(B4,S2) strongly convergent to a map u in W1,3(B4,S2) one again poses the question about the limit of the topological singularities Singtopun. Recall that the homotopy class in π3(S2) of a regular map ψ :S3 →S2 is given by the Hopf degree of ψ, which is topologically the linking number of the inverse images of two regular values of ψ and is analytically given by the integral
Hopf degree(ψ) = 1 4π2
η∧ψ#ωS2
where η is any 1-form on S3 verifying dη =ψ#ωS2. A simple integration by parts then shows us that, similar to the case of R∞,2(B3,S2), the topological singularity of the map un∈ R∞,3(B4,S2) may be written
Singtopun =
a∈An
mn,a[[a]] = ∗d
ηn∧u#n ωS2
2π
where ηn is any 1-form on B4\ An verifying dηn = u#nωS2. Our principal preoccupation is then to study a possible convergence of∗dηn∧u#nωS2
and for example to verify whether, as in the case of W1,2(B3,S2), or not there exists a sequence of 1 dimensional currents In having∂In=a∈Anmn,a[[a]] and having uniformly bounded masses (i.e. a∈Anmn,a[[a]]W−1,1 ≤ C independent of n). However, such an attempt runs into the basic problem of the actualfailure of suitable bounds for these convergences.
To see this failure, one starts withu∈W1,3(B4,S2)and first proves without difficulty that du#ωS2 = 0 and that the 1-form η verifying dη = u#ωS2 of
“maximal” regularity is a priori the Coulomb gauge which is the solution of dη =u#ωS2 in D2(B4)
d∗η =0 in D1(B4) ι#∂B4η =0
where ι∂B4 is the inclusion of ∂B4 into R4. Since the form u#ωS2 is only in L3/2(B4), the solution η of this problem is in L12/5(B4) ⊃⊃ L3(B4), thus a priori η∧u# is not in L1loc(B4), and it seems difficult to give this a meaning even in D(B4). In [R1] the second author showed in fact that the above small calculation is optimal in establishing that
(0.2) log inf
S3|ψ|3dH3 : ψ :S3→S3, Hopf degree(ψ)=d
≈ 3 4log d asd → ∞. This 3/4, which replaces the 1 that occurs in minimizing p-energy among (topological) degreed maps from Sp to Sp, appears when one expresses the Hopf degree by means of the above Coulomb gauge. One shows that it is optimal by using maps whose inverse images are self-linked (see [R1]). This is the source of all the difficulties encountered below in this paper. Using the argument of Section 2.5, this 3/4 estimate allows us to construct a sequence un∈R∞,3(B4,S2) such that
un → u strongly in W1,3 but
inf {M(In):∂In=Singtopun} → +∞,
and one does not see a priori how Singtopun may converge in D(B4). It is necessary to envision some convergences in larger spaces for some objects whose masses may tend to infinity.
Introduction of “Scans”.
In face of the impossibility of getting convergence in D(B4) of our topo- logical singularities of maps un ∈ R∞,3(B4,S2) strongly convergent to a u ∈ W1,3(B4,S2), we will adapt the approach that Giaquinta, Modica, and Souˇcek [GMS1], [GMS2] used for the case πp(Sp), and we will be interested in a possible convergence of a sequence of graphs of smooth maps in C∞(B4,S2).
Therefore let un∈C∞(B4,S2) converge W1,3 weakly to u∈W1,3(B4,S2). One will suppose for simplicity that un andu are constant on ∂B4 (see Section 2.3). It is not difficult to see that, for all u ∈W1,3(B4,S2), the graph of u is a rectifiable current satisfying
∂Graph(u)=0 in B4.
In fact u may be approximated strongly by a map v∈R∞,3(B4,S2) and the 3 dimensional flat current∂Graph(v), being supported in B4 in the 2 dimensional set sing(v)×S2, must vanish [F], 4.1.21.
The boundary of the graph thus does not characterize, in this case, the failure of the strong approximability by smooth maps. On the other hand, one can prove, from the vanishing of π1
B4\Singtop(v), the existence of a Hopf lifting v˜ of v for the Hopf map :S3→S2 (i.e. v˜:B4→S3 and ◦ ˜v=v).
For such a v one has that
∂Graph(v)˜ =Singtopv×[[S3]]
so that the boundary of the graphs of Hopf lifts do characterize the topological singularities. One is therefore led to take a smooth Hopf lifting u˜n of the map un and study the possible convergence of Graphu˜n to a limit object in the form
“Graphu˜ +I ×[[S3]]” where I will be a “reasonable” object connecting the topological singularities of u. There exist a lifting operation for the Hopf fibra- tion which is associated (Section 2.1) with the extraction of the Coulomb gauge described above. Let u˜n denote such a Coulomb lift for which one then has control in W1,5/12 but not inW1,3as seen by the example of Section 2.5. While Example 2.5 shows the possibility that MGraph(u˜n) → ∞, we nevertheless establish in Section 2.4 an L3/4 bound for the mass of hyperplanar slices.
More precisely, for each unit vector v ∈ S3 and t ∈ R, we have the corresponding hyperplane h(v,t)= {x ∈R4:× ·v=t} oriented by the normal vector v. Intersecting the 4 dimensional current Graph(u˜n) by h(v,t)×[[S3]], or equivalently slicing [F], 4.3, by the projection (x,y) → x ·v, gives the 3 dimensional current Graphu˜n|h(v,t) corresponding to restricting u˜n to the hyperplane h(v,t). We show
(0.1) sup
v∈S3
1
−1
M3/4
Graphu˜n|h(v,t)∩B4
dt ≤ C
1+
B4|∇u|3d x
.
Such control on the integral of the masses of the slices to a power less than 1 suggests characterizing an object as a collection of all (or almost all) its slices.
This is the notion of a scan which we will define below.
As motivation, consider the following simplified problem where one is given a sequence of unions of immersed oriented closed curves n= ∪k k
n in the closed unit ball in R2 satisfying the bound
(0.2) sup
v∈S1
1
−1
Cardα n∩L(v,t)dt ≤ C independent ofn.
where L(v,t) denotes the line {x ∈R2 : x·v=t}. If α=1, then this bound gives us control on the mass (or total length) of the 1 dimensional current n, independent of n. Knowing that ∂ n=0, one is then in position to apply the Compactness theorem of Federer-Fleming and deduce that, after passing to a subsequence, the n converge to a limit rectifiable current .
When α <1, (0.2) does not guarantee control of the total length M( n), and there is no reasoning that allows us to deduce some convergence of the
n as distributions. One thus introduces a map µn from the space of oriented lines S1×R to the space M of atomic measures on R2 which at almost every (v,t) associates the 0 dimensional intersection current
µn(v,t)= n∩L(v,t)
which is a sum of point masses with integer multiplicities. Being given a reference frame {e1,e2} of R2, one equips M with the following metric
d(µ, µ)=inf
Mα(S)+ 2
j=1
1
−1Mα
n∩L(ej,s)ds :µ−µ =S+∂T
, and one verifies that the above µn is a measurable function from S1×R to M equipped with the topology induced from the metric d.
The current equation ∂ n=0 translates to a new boundary zero condition for the corresponding scanµn (see Section 1) which is a compatibility condition allowing one to see that µn is the scan of an underlying closed object in the plane. Also estimate (0.2) and this boundary zero condition imply the following regularity estimate:
dµn(v,t), µn(v,t ) ≤ Fn(t)|t −t |α for all v∈S1 and some Fn in L1/α(R) weak = L1/α,∞ with
FnL1/α,∞≤ C sup
v∈S1
1
−1
Mα
n∩L(v,s)ds.
Such uniform control of this regularity permits us then to establish, after passing to a subsequence, convergence a.e. of µn to a scan limit µ, a limiting object at least which, though a priori strange, is convenient to study in the particular cases we consider. When α=1, we recover the characterization by Ambrosio and Kirchheim of rectifiable objects by means of a weakly BV maps with values in metric spaces. See also White’s rectifiability proof [W3].
Returning now to the original problem of the sequence un of maps in C∞(B4,S2) converging weakly in W1,3 to u. To each un one may associate the scan of its Coulomb lift
Gu˜n :S2×R → R3(B4×S2) , Gu˜n(v,t)=Graphu˜n|h(v,t), the space R3(B4×S2)denoting the 3 dimensional rectifiable currents in B4×S2. On R3(B4×S2) one considers the distance
de(P,Q)=inf
M(S)+ 4
j=1
M
T∩h(ej,t)34dt : P−Q =S+∂T
where e=(e1,e2,e3,e4) is a fixed frame of R4. This time the control of (0.1) translates to a regularity for the scan Gu˜n:
dGu˜n(v,t),Gu˜n(v,t ) ≤ Fn(t)|t −t |34 for all v∈S3 and some Fn in L4/3,∞ with
FnL4/3,∞ ≤ C
1+
B4|∇un|3d x
.
Rather than referring to general properties of the space L4/3,∞, we prove in Section 9, for the reader’s convenience, the appropriate precise compactness statement needed. With this, we then establish the following result which is the analogue for W1,3(B4,S2) of Theorem 0.6.
Theorems 6.1, 7.2.Suppose that un∈C∞(B4,S2)converge weakly in W1,3to u. Then, after passing to a subsequence, one has the convergence almost everywhere of scans of Coulomb lifts
Gu˜n → Gu˜+I ×[[S3]]
where I×[[S3]]is the scan of a rectifiable set R×S3inB4×S3equipped with an integer multiplicityθ, measurable on , such that
R|θ|34 dH1 < ∞.
In the sense of scans,
∂Gu˜ + I ×[[S3]]=0 in B4.
While it is still unknown whether an arbitrary map u ∈ W1,3(B4,S2) is such a weak limit of smooth maps, we can nevertheless still use the scan Gu˜
of the graph of its Coulomb lift to express the strong approximability criterium Lemma 2.7.
u∈C∞(B4,S2)W1,3 ⇐⇒ ∂Gu˜ =0 inB4.
In any case, this scan boundary may again be capped off by a vertical scan:
Theorem 8.1.If u is any element of the Sobolev space W1,3(B4,S2), then
∂Gu˜+I ×[[S3]]=0 in B4 where, for allv∈S3and a.e. t∈R,
I ×[[S3]]h(v,t)=
a∈Av,t
mv,t[[a]]×[[S3]], for some finite subset Av,t of h(v,t)and non-zero integersmv,twith
R
a∈Av,t
mv,t
3 4
dt ≤C
1+
|∇u|3d x
.
We can use this estimate to show only that the I of Theorem 8.1 is carried by a set of finite H4/3 measure, and not, as in the case of Theorem 7.2, carried by a 1 rectifiable set. In fact the optimal structure of such an I seems related to the question of the weak sequential density of C∞(B4,S2) in W1,3(B4,S2). For general Sobolev spaces of mappings, strong approximability by smooth maps has been well-studied (see [Be1], [HgL1], [HgL2]), but the same problems for the weak topology are still largely open. (see [PR]).
As we have argued here and in the seminar [HR1], the scans defined in this work allow one to study Sobolev mappings via their graphs by exploiting esti- mates valid on restriction to hyperplanar subspaces. Approximation properties characterized by restricting to lower dimensional subspaces also occurs in the work [M] of Mucci. Our limiting objects however are no longer currents, and the scans we introduce thus strictly extend and generalize the Cartesian currents of [GMS2]. General rectifiable currents (not necessarily related to smooth map- pings) have also been understood and well-studied through slicing [AK], [W3].
So general scans (as in the motivating example (0.2)) should provide a useful extension of various classes of currents. The second author and T. DePauw
[HD] have studied some compactness, rectifiability, and variational problems for such scans.
Recently [HR3] we have also realized another argument for W1,3(B4,S2) for producing a suitable connecting “bubbled” scan I which avoids the use of Hopf liftings. This approach allows, for an arbitrary manifold N, identification of suitable connections for those topological singularities that issue from the infinite part ofπp(N), πp(N)⊗Q. In [HR3] we study certain Gauss integrals in product spaces and the Novikov integral expressions [Nov] of rational homotopy.
The use of scans again seems necessary to expedite such an approach. One expects, in fact, because of considerations which led to the above 3/4 and to the exponents of Gromov [Gr], that this power should be replaced by other powers strictly less than 1 (except in the simple case πp(Sp) described above) and that therefore the masses of these connections I should again be infinite.
The torsion part of homotopy groups may also contribute to bubbling, failure of strong density, topological singularity, etc. Much less is known. Pakzad and Rivi´ere [PR] studied π1(RP2) ≈ Z2 and more generally πm−1
(m −2)−
connected N. In [HR2] we analyze the effect of π4(S3)≈Z2 on the second order Sobolev space W2,2(B5,S3).
1. – Hyperplanes inR4and scans
We identify S3×R with the space H of oriented hyperplanes in R4 by associating with each pair (v,t)∈S3×R the hyperplane
h(v,t)≡ {x∈R4:x·v=t}
oriented by the normal vector v. Thus H is equipped with the standard metric of S3×R and the 4 dimensional Hausdorff measure (H3|S3)×H1, that is, dh(v,t)=dH3vdt.
We also occasionally leth=h(v,t)denote the corresponding 3 dimensional current (See [F], [S], [GMS2] for notations.) that is the boundary of the standardly-oriented half-space,
h=∂[[{x∈R4:x·v <t }]].
The orientation is described by either the constant tangent 3 vector h or by the dual normal 1 vector h∗=v.
We will study a smooth map w ∈C∞(R4,S3) in terms of its restrictions to hyperplanes. In particular, for any h∈ H, we consider the oriented graph of wrestricted to H as the 3 dimensional current
Gw#h= where Gw(x)=x, w(x).
Thus, Gw#h∈R3,loc where we here use the abbreviations Ri =Ri(R4×S3) , Ri,loc=Ri,loc(R4×S3) ,
for the groups of i dimensional integer-multiplicity rectifiable and locally recti- fiable currents in R4×S3 ([F], [S], [GMS2]).
We also need various projections:
p: R4×S3 → R4, p(x,y) = x, q: R4×S3 → S3, q(x,y) = y,
πv : R4 → R, πv(x) = v·x, pv =πv ◦p : R4×S3 → R, pv(x,y) = v·x, for x ∈R4, y∈S3, and v∈S3.
In terms of boundary or slicing (see [F], 4.3 or [S],),
Gw#h(v,t)=∂Gw#[[πv−1(−∞,t)]]=<Gw#[[R4]], pv,t > . Note that
∂Gw#h =Gw#∂h=0, p#Gw#h=h, q#Gw#h=w#h.
Moreover, for any two hyperplanes h, h ∈ H, we have the compatibility property that
Gw#h∩h ×[[S3]]=Gw#h∩h×[[S3]]
because
Gw#h(v,t)∩h(v,t)×[[S3]] =<<Gw#[[R4]], pv, t >, pv ,t >
=<<Gw#[[R4]], pv,t >, pv, t >
=Gw#h(v,t )∩h(v,t)×[[S3]]. In general, we define a scan to be any function
S : H → R3
satisfying
S(h)∩h ×[[S3]] = S(h)∩h×[[S3]]
for almost every pair h, h ∈ H. The special scan S = Gw# is called the scan of the map w∈C∞(B4,S3). More generally, for any current T ∈R4 with p#T = [[R4]], there is an associated scan, ScanT, defined by the hyperpla- nar intersection
(ScanT)(h)≡T ∩h×[[S3]] fora.e. h∈H,
so that, in terms of slicing, (ScanT)h(v,t))=<T,pv,t > for a.e. t ∈R.
Thus a scan may be considered as a generalization of a Cartesian current ([GMS2]).
The compatibility condition indicates that scans may be determined by their values on a smaller family of hyperplanes, for example, the coordinate hyper- planes associated with some orthonormal frame. This will be first illustrated in Section 6 where use of standard coordinate hyperplanes will be sufficient to establish the convergence of a sequence of scans of smooth maps. Actually, our limiting scan will only be determined at almost every h ∈ H, but will nevertheless inherit some properties from the scans of rectifiable currents.
In particular, we may define the notion of a scan cycle, that is, what it means for a general scan to havezero boundary. For the scan of a smooth map w∈C∞(R4,S3) and any subset U of R4 of locally finite perimeter
∂Gw#∂[[U]]=Gw#∂∂[[U]]=0 and Gw#∂[[U]]q#ωS3
=Gw#[[U]]q#dωS3
=0. A definition suitable for general scans may be made by using polyhedral do- mains.
A polyhedral frontier is a current ∂[[U]] where U is an open polyhedral domain in R4. For a polyhedral domain U with k distinct 3-dimensional faces, we may represent
∂[[U]]= k
i=1
hi∂U
where each hi is supported by the hyperplane containing some 3-face of ∂U and hi∗
is the outward unit normal of this face. We now say that
∂S=0
(or that S is a span cycle) if, for almost all polyhedral frontiers ∂[[U]] = k
i=1hi∂U as above, the rectifiable current S∂U ≡
k i=1
S(hi)p−1(∂U) satisfies the two conditions
∂S∂U
=0 and S∂U
q#ωS3
=0.
Here, “almost all” means that an exceptional set Z of polyhedral frontiers has measure zero in the sense that
{(h1,h2, . . . ,hn)∈Hn : ∂[[U]]∈ Z for some component U of R4\ ∪ni=1hi} has measure zero in Hn for all n. The necessity of the second condition in the definition is shown by the oriented graph of x/|x|, which is a current in R4 with nonzero boundary [[0]]×[[S3]] whose corresponding scan satisfies the first, but not the second condition. In fact, as we will see later, the graph of any map in W1,3(R4,S3) satisfies the first condition. For the scan of a rectifiable current, we have the following:
Lemma 1.1. If T ∈ R4,loc andM(∂T) < ∞, then ∂T = 0 if and only if
∂(ScanT)=0.
Proof. For almost all polyhedral domains U as above, [F], 4.3 gives the formula
(ScanT)∂U = ∂Tp−1(U) − ∂Tp−1(U)) . Thus if ∂T =0 , then ∂(ScanT)∂U
=∂∂Tp−1(U)=0 and
(ScanT)∂U
q#ωS3=∂Tp−1(U)q#ωS3=Tp−1(U)q#dωS3=0.
Conversely, suppose ∂(ScanT)=0. Then, for almost all h(v,t)∈H,
< ∂T,pv,t >=0
because, we may, for any form φ ∈ D2(R4×S3), choose a large polyhedral domain U with ∂U ∩sptφ = h(v,t)∩sptφ, hence, < ∂T,pv,t > (φ) =
∂(ScanT)∂U
(φ)=0.
It follows that, for ∂T almost all points z, the approximate tangent 3 plane Lz associated with −→∂T(z) has p(Lz)=0. In fact, if p(Lz) contained a line, then pv|Lz would have rank one for a.e. v∈S3. By [F], 4.3, this would give
z ∈ spt< ∂T,pv,pv(z) >
for ∂T almost all such z, contradicting the vanishing of < ∂T,pv,t > for a.e. t.
Thus −→∂T(z)= ±0,−−→
[[S3]]q(z) for ∂T almost all z, and, since∂∂T = 0, an elementary argument [H], Th.1, shows that
∂T =
a∈A
ma[[a]]×[[S3]]
for some finite subset A of R4 and some integers ma. Using now the second condition of∂ScanT)=0 with almost any polyhedral domain U withU∩A= {a} =U∩ A, we deduce that
ma=(∂T)p−1(U)q#ωS3
=(ScanT)∂U + ∂(Tp−1(U)q#ωS3
=0+0.
Thus ∂T =0.
2. – The Hopf map and Coulomb lifting Recall that the Hopf map
: S3 → S2
may be described explicitly by the formula (z, w)=z/ w where we identify the domain S3 with
{(x1+ix2,x3+ix4)∈C2 : |x1|2+ |x2|2+ |x3|2+ |x4|2=1}
and the rangeS2 with the extended complex planeCˆ via the usual stereographic projection. One readily checks that
(z, w)=(z, w)if and only if (z, w)=eiθ(z, w)for someθ ∈R. Also, pulling back the volume form ωS2 via gives
#ωS2 =4(d x1d x2 + d x3d x4)=2dα where α=x1d x2 − x2d x1 + x3d x4 − x4d x3.
Let M =S3, R3,R4 (or any oriented simply 2-connected Riemannian man- ifold). For any smooth map u:M →S2, a smooth map uˆ : M →S3 satisfying
◦ ˆu=u
is called a Hopf lift of u and a smooth 1 form η on M satisfying dη=u#ωS2
is called a gauge for u. For a Hopf lift uˆ of u, the formula η=2uˆ#α
clearly defines a gauge for u. Conversely.
Lemma 2.1. Any gauge η for u ∈ C∞(M,S2) equals2uˆ#α for some Hopf liftu of u. The liftˆ u forˆ ηis unique up to multiplication by eiθ for some constant θ ∈R, and
|∇ ˆu|2= 1
4|η|2 + |∇u|2.
Proof.First note that the u pull-back of the bundle :S3→S2is a trivial S1 bundle over M. Using a trivializing map, we readily find some smooth Hopf lift uˇ of u. Since
d2uˇ#α−η=0,
